Math. Anal. Appl. 370 (2010) 703-715 Contents lists available at

J. Math. Anal. Appl. 370 (2010) 703–715
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa

An abstract Nyquist criterion containing old and new results
Amol Sasane
Optimization and Systems Theory Division, Mathematics Department, Royal Institute of Technology (KTH), 100 44 Stockholm, Sweden

a r t i c l e i n f o a b s t r a c t

Article history:
Received 8 December 2009 Submitted by S. Power
Keywords:
Nyquist criterion Control theory Banach algebras

1. Introduction


We prove an abstract Nyquist criterion in a general set up. As applications, we recover various versions of the Nyquist criterion, some of which are new.
2010 Elsevier Inc. All rights reserved.

Harry Nyquist, in his fundamental paper [19], gave a criterion for the stability of a feedback system, which is one of the basic tools in the frequency domain approach to feedback control. This test, which is expressed in terms of the winding number around zero of a certain curve in the complex plane, is well known for finite dimensional systems; see for example [25] or Theorem 5.2 in this article. There are several extensions of this test for other classes of systems as well; see for example [3,5,6]. Thus the problem of obtaining a Nyquist criterion encompassing the different transfer function classes of systems is a natural one; see [15], [21, p. 65].

In this article, we will prove an “abstract Nyquist theorem”, where we only start with a commutative ring R (thought of as the class of stable transfer functions of a linear control system) possessing certain properties, and then give a criterion for the stability of a closed loop feedback system formed by a plant and a controller (which have transfer functions that are matrices with entries from the field of fractions of R). We then specialize R to several classes of stable transfer functions and obtain various versions of the Nyquist criterion. In the section on applications, we have given references to the known results; all other results seem to be new.
The article is organized as follows:
(1) In Section 2, we describe the basic objects in our abstract set up in which we will prove our abstract Nyquist criterion. The starting point will be a commutative ring R. We will also give a systematic procedure to build the other basic objects starting from R in the case when R is a Banach algebra.
(2) In Section 3, we will recall the standard definitions from the factorization approach to feedback control theory.
(3) In Section 4, we prove our main result, the abstract Nyquist criterion, in Theorem 4.1.
(4) Finally in various subsections of Section 5, we recover some old versions of the Nyquist criterion as well as obtain new
ones, as special instances of our abstract Nyquist criterion.

E-mail address: [e-mail].
0022-247X/$ – see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2010.05.028

704 A. Sasane / J. Math. Anal. Appl. 370 (2010) 703–715